Call a person's $j$-th dance a dance of level $j$.

Pairing a man's level $p$ with a woman's level $q$ is legal exactly when $p\ge q$. Before that turn, their counts are $p-1$ and $q-1$.

Now imagine fixing final counts $x_i$ for the men and $y_i$ for the women. Both totals must equal the number of turns $T$.

Each man contributes levels $1,2,\ldots,x_i$; each woman contributes levels $1,2,\ldots,y_i$.

The task becomes matching these two collections so that every male level is at least the female level matched to it. This removes the need to reason about every possible schedule.